Study guide and 3 practice problems on:

## Fundamental theorem of line integrals

If $\mathbf{F}(x,y) = \nabla \phi(x,y)$ and $C$ is any curve going from $(x_0, y_0)$ to $(x_1, y_1)$: $$\int_C \mathbf{F}\cdot d\mathbf{r} = \phi(x_1, y_1) - \phi(x_0, y_0).$$
The fundamental theorem applies if the vector field $\mathbf{F}(x,y)$ is conservative. This is, when $\mathbf{F}(x,y) = \nabla \phi(x,y)$ for some $\phi$.